with m for 0 and h for 1, the inverse binary expansion of n, without the leading 1, gives the sequence of means. (End)

As function of the absolute value, defines the minimal Euclidean function v on Z\{0}. A ring R is Euclidean if for some function v : R\{0}->N a division by nonzero b can be defined with remainder r satisfying either r=0 or v(r)<v(b). For the integers taking v(n)=|n| works, but v(n)=floor(log_2(|n|)) works as well; moreover it is the possibility with smallest possible values. For division by b>0 one can always choose |r|<=floor(b/2); this sequence satisfies a(1)=0 and recursively a(n)=1+max(a(1),...,a(floor(n/2))) for n>1. - Marc A. A. van Leeuwen, Feb 16 2011

Maximum number of guesses required to find any k in a range of 1..n, with 'higher', 'lower' and 'correct' as answers. - Jon Perry, Nov 02 2013